How can Earth get nuked the way it did in The Expanse?

There are probably two main reasons:

1. Technical

It is very, very hard to shoot something relatively small moving very, very fast, using state of the art stealth technology. To make things even more difficult, the Mars has put the missiles in a close proximity to Earth, so there is a very little time to guide the counter-missile against the attacker before the missile splits into separate warheads.And, last but not least, effects of the counter-missiles explosions is very diminished in void - even with a nuclear warhead, you have to hit pretty close to target (you can read NASA whitepaper about it).

Please also note, that Mars has a significant technological advantage over Earth, which Earth only manages to balance by the number of the raw resources - in the "Epstein" short story in the universe, someone compares Mars and Earth to the situation with Germany and USSR during WW II: German tanks were much better, but Russians have much more of them, Mars has better nukes, ships and marines but Earth has so much more of them. This is the reason that the tentative balance is being held in check.

It was like a nuclear attack from a submarine off the coast - created exactly to give the opponent as little warning as possible. Shooting multiple counter-missiles could help... probably, if we assume that they won't start interfering with one another. But that leads to another issue:

2. Politics

If you start stockpiling missiles, your opponents might get nervous and do the same, creating this way an arms race which no one really wants (besides its obvious danger it is also very costly). You could argue that you are putting a defensive counter-missiles just for "your own protection", but that looks like a blatant lie, because history has displayed many, many times that increasing your protective armament was often a prelude to an offensive campaign and because your opponent might start thinking that you are stockpiling an offensive weapons too.

So what is the best solution? Mutually Assured Destruction - yes, the best protection against enemy's nuclear missiles are your own nukes pointed at them. If both sides know that they will be completely obliterated in case of war they won't want to go to war. You don't have to develop expensive (and dubiously effective) counter-weapons and you don't need to rush into an open arms race.

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In the comment, the OP mentioned that:

we see ships can shoot down incoming torpedoes -- torpedoes that have often burned hundreds of thousands of kilometers thus accruing incredible speed -- with guns. This is much harder than detonating a nuke next to the incoming missile

I need to point the difference between "torpedoes" and ballistic missiles and between space ships and planet Earth:

• Even very advanced torpedo will be much simpler devices that hyper-advanced first strike nuclear missile, with the former being not much more than self-guiding "bullet", the latter would be more akin to an autonomous space ship with its own countermeasures against incoming fire. The reason for that is the cost of failing to hit: if a torpedo fails to connect, you can send another one and at the worst case, the attacking ship is destroyed by the returned fire. If the first strike nuclear missile fails to destroy your opponent's ability to return fire, you might lose a whole planet in the returning nuclear strike.
• While we don't know the exact size of the space ships in the show, we can roughly guess that Rocinante is probably much less than 100 m long and the biggest combat ships would have maybe a few kilometres in length. You can easily cover such area with defensive, anti-torpedoes weaponry so your whole ship is protected. In contrast, you can't really do it with the whole planet that has so much bigger surface to protect leaving much bigger gaps through which your enemy can slip.

TL;dr

it is much harder to destroy an advanced ballistic missile that has been designed to slip through your defenses than a simple ship-to-ship torpedo, especially when you can't build as much defensive structures that you'd like.